Moduli spaces of quadratic rational maps with a marked periodic point of small order

The surface corresponding to the moduli space of quadratic endomorphisms of P1 with a marked periodic point of order n is studied. It is shown that the surface is rational over Q when n 5 and is of general type for n = 6. An explicit description of the n = 6 surface lets us find several infinite families of quadratic endomorphisms f : P1-> P1 defined over Q with a rational periodic point of order 6. In one of these families, f also has a rational fixed point, for a total of at least 7 periodic and 7 preperiodic points. This is in contrast with the polynomial case, where it is conjectured that no polynomial endomorphism defined over Q admits rational periodic points of order n > 3.